Mild solution to parabolic Anderson model in Gaussian and Poisson potential

Abstract

We consider the parabolic Anderson model in a special random homogeneous potential that is generated by the Gaussian and Poisson random medium together. In our model, the coefficient V(x) is not Hölder continuous with positive probability, and thus the model is unlikely to have a path-wise solution. We construct a mild solution for the parabolic Anderson model with random potential of the form \documentclass[12pt]{minimal}\begin{document}$-V(x)=-\int _{\mathbb {R}^d}K(y-x)[\omega ({\rm d}y)+W({\rm d}y)]$\end{document}−V(x)=−∫RdK(y−x)[ω(dy)+W(dy)], where ω and W denote the independent standard Poisson point process and centred Gaussian field, respectively. The case where the potential switches in sign and the Poisson field is absent is handled as well.